1974 AHSME Problems/Problem 20

Problem

Let

\[T=\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}.\] (Error making remote request. Unexpected URL sent back)

Then

$\mathrm{(A)\ } T<1 \qquad \mathrm{(B) \ }T=1 \qquad \mathrm{(C) \ } 1<T<2 \qquad \mathrm{(D) \ } T>2 \qquad$

$\mathrm{(E) \ }T=\frac{1}{(3-\sqrt{8})(\sqrt{8}-\sqrt{7})(\sqrt{7}-\sqrt{6})(\sqrt{6}-\sqrt{5})(\sqrt{5}-2)}$

Solution

Let's try to rationalize $\frac{1}{\sqrt{n+1}-\sqrt{n}}$ . Multiplying the numerator and denominator by $\sqrt{n+1}+\sqrt{n}$ gives us $\frac{1}{\sqrt{n+1}-\sqrt{n}}=\sqrt{n+1}+\sqrt{n}$ .

Therefore,

$T=(3+\sqrt{8})-(\sqrt{8}+\sqrt{7})+(\sqrt{7}+\sqrt{6})-(\sqrt{6}+\sqrt{5})+(\sqrt{5}+2)=5.$

Hence the answer is $\boxed{\text{D}}$ .

1974 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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1974 AHSME Problems/Problem 20

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